3.032 Class Questions Re: Quiz 2 Topics
Fall 2007
Note: These are responses to class members’ questions in order of emails received, not necessarily importance.
My responses are intended to supplement your class lecture notes and reading material.
1. Please go through the shrimp at the base of the ocean problem again. I am not clear the difference
between stress and shear when it comes to the death of the shrimp. That was a great example you brought
up.
A material or an animal within the ocean is subjected to pure compressive hydrostatic stress (normal stresses in x,
y, and z are equal in magnitude). The maximum normal stress criterion for plastic deformation predicts that a
material will yield if the normal stress reaches the value obtained when the material fails under uniaxial
tension/compression. In other words, the material will fail if ANY of the normal stresses reaches yield strength sy.
The uniaxial compressive stress required to yield or kill a shrimp (0.1 MPa), so this criterion predicts that a shrimp
would yield/die if it reached a depth in the ocean where the pressure reached 0.1 MPa. However, shrimp live
happily at depths where pressure is 10 MPa, so this criterion does not accurately capture the “yield criterion” that
predicts plastic failure of shrimp. We discussed that the reason for this failure is that many materials including the
biopolymers that make up a shrimp fail not due to high normal stresses but to high shear stresses.
From Mohr’s circle, it is easy to see that hydrostatic stress fails to generate shear stress on any plane inside the
material: this is a principal stress state where all 3 normal stresses are located at the same point on the σ-axis, so
you cannot draw a circle to define planes on which the shear stress is nonzero.
2. Principle [sic] stress, principle [sic] shear: What does this mean in an engineering system? If I have a
ceramic I know has a yield stress of x MPa, how does a 2 x 2 matrix of stress and shear indicate how and in
what orientation I can implement it?
Engineers need to be able to determine the principal state of stress or of strain in a material to identify the
maximum/minimum values that normal/shear stresses/strains will attain inside the material. As we learn from the
yield criteria, if these maximum values reach the magnitudes required to yield the material, the material will deform
permanently (plastically) rather than reversibly (elastically). This often leads to functional failure of the structure or
device, and computation of the principal stresses corresponding to a triaxial stress state is the fastest way to
determine whether yielding will occur according to these criteria.
By “2 x 2 matrix of stress and shear”, I think you mean a matrix σij that includes both normal and shear stresses; if it
is a 2x2 matrix then you are assuming plane stress: the only nonzero normal/shear stresses are contained in one
plane (say, x-y plane), and there are no normal/shear stresses in the orthogonal plane of the material (say, the zplane). This is a reasonable approximation for thin sheets of material, where z is the through-thickness direction.
If you had a ceramic of known yield stress x MPa and knew your application would place this ceramic under this
plane stress state, you would want to immediately determine the principal stress state. Why? First, you could then
determine whether these stresses are sufficient for yielding to occur, according to your choice of yield criteria.
Second, this would tell you the orientation and magnitude of the maximum normal stress. Although we’ve not yet
covered fracture in 3.032 yet, ceramics are typically brittle materials that fracture rather than plastically deform, and
the maximum normal stress criterion does a good job at predicting the fracture stress of such materials. If the
maximum normal stress in the ceramic reaches the fracture stress of your ceramic, you know right away that the
ceramic device will fail and you would choose another ceramic of larger fracture stress.
If the maximum normal stress is below the fracture stress of your ceramic, you’d still want to consider modifying the
ceramic via processing to either increase its fracture stress or adding fibers that would be aligned to block growth of
cracks. Thinking back to the pressure vessels/hot dogs, you can intuit that the cracks would run perpendicular to
the maximum normal stress, and you would then align these fibers parallel to the maximum normal stress
orientation (the principal stress axis orientation) to be sure you’d block such cracks.
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3. Do components like fibers on a silicon substrate really undergo stress tensors like in that problem set?
How? Do engineers at intel and other device companies really look at stress states like that?
Yes, engineers at electronic device companies like Intel in fact consider such stress states on silicon. Typically,
such engineers are concerned with triaxial thermal stresses when these devices reversibly heat and cool. If the
silicon substrate is both heated and loaded mechanically, which is common in such devices, both normal and shear
stresses are generated within the Si wafer. Interconnect lines of metal and device insulators of oxides are so thin on
such devices that, at the interface where those tiny lines of materials join to the silicon substrate, the strains
generated in those small-volume “fibers” or films are exactly equal to the strains in the Si.
4. Why do I want to rotate a stress state x degrees? What does this physically mean in an engineering
system?
We’ve discussed this through several different examples in class and recitation, as well as in Lab 1. We are not
rotating a stress state, but rather resolving the stresses or transforming the stresses onto a new set of coordinate
axes. Coordinate axes are arbitrary choices to analyze a real engineering system; we often pick axes that align with
the shape of the object or the orientation of the applied stresses/strains, but that is purely arbitrary. If we were to
draw a different set of coordinate axes rotated by some angle theta, we’d come up with a different combination of
stresses/strains.
This is the same exercise as resolving a force vector that is at some arbitrary angle into its “vertical” and “horizontal”
components. We have not rotated the force, just expressed that force in terms of a new set of axes. This makes it
easier for engineers to analyze all the complex forces acting on a device, e.g., by balancing all the vertical and
horizontal components to make sure the device will not move.
In the case of stresses inside a material, we resolve stresses onto coordinate axes to easily identify planes inside
the material that are under high normal stress/strains or high shear stresses/strains. These are locations that we
identify with respect to our initial set of axes, so we say that these orientations of high normal or shear stress/strain
are “rotated some angle theta with respect to our initial axes.” As a materials engineer, you know that materials will
deform and/or fail under sufficiently high stress states, so you need to know the orientation of regions under high
stress (likely locations of failure). You also need to know how a stress state is distributed onto planes defined by the
unit cell planes of a crystal; or onto planes that define the orientation of precipitates, particles, grain boundaries, or
fibers within the material.
5. Could you go over the dislocation stress fields and also the force vs. extension relationships for the
three types of yield conditions.
(a) Yield conditions (covered first, as this is a continuum concept applicable to crystalline and noncrystalline
materials):
1. Maximum normal stress criterion for yielding: material yields when any of the three principal normal stresses
reach the normal stress attained when the material yields under unaxial loading, which is exactly the yield stress.
2. Maximum shear stress criterion/Tresca: material yields when the shear stress reaches the maximum shear stress
attained when the material yields under unaxial loading. From Mohr’s circle, you can see that the shear stress
attained upon yielding under uniaxial loading is (σ1 – σ3)/2 = σy/2.
3. Maximum distortional energy/von Mises: material yields when the effective stress reaches the effective stress
when the material yields under uniaxial loading. The effective stress is defined by the 2nd invariant of the deviatoric
stress tensor, and can be calculated from any stress tensor σij or the principal stresses σ1, σ2, and σ3 corresponding
to that stress tensor.
(b) Dislocation stress fields: The stress generated inside the crystal due to the presence of a single screw or edge
dislocation can be calculated from the displacement the dislocation creates: u Æ ε Æ σ in our usual way, where the
arrows here indicate a partial derivative and a constitutive relation, respectively. We’ll cover this more extensively in
the elective, 3.14. In this class, you should understand how these stress fields are derived (just as we derive stress
in a material from any displacement applied to that material), and how these dislocation energies and stresses
depend on elastic properties (G and ν) and length scales (b) of the crystal.
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6. It would be nice to clarify the different types of elasticity we learned and the important differences
between them, and what materials they are relevant for.
We’ll do this in the quiz review sessions together. You should try to make a table of this for your crib sheet, using
your notes and course reader.
7. Why do only gases readily obey Newton's viscosity law but not high molecular weight polymers?
When the motion of particles is uncorrelated, Newton’s law of viscosity holds well: the resistance to shear flow or
the shear strain rate varies linearly with the applied stress. For gases and low molecular weight polymers or fluids
of low viscosity, the number of particles/chains per unit volume is so low that the motion of one particle/chain is
really independent of the motion of the nearby particles/chains. As the molecular weight of a polymer increases, the
chains are increasingly entangled and thus the motion of one chain depends in part on the motion of the nearby
chains.
8. Overview of the graphs for the LVE solid. Specifically, why does the KV model fail to describe relaxation
in polymers? why does the Maxwell model fail to accurately predict creep in solid polymers? (You gave an
excellent explanation in class during those lectures but I can't remember fully:)
“Why” these models describe only creep or only relaxation well is attributed to what kind of material for which they
were originally designed to match the experimental data. The Maxwell model describes a spring and dashpot in
series, and was proposed to explain the time-dependent deformation of very viscous fluids like tar. If this model is
subjected to an immediate, constant tensile stress (tensile creep), the spring can extend instantaneously to a strain
ε = σ/E; a real polymer does not deform instantaneously, but in fact exhibits a lagged increase in strain. The K-V
model describes a spring and dashpot in parallel. If this model is subjected to an immediate, constant strain e,
relaxation is the decrease in stress inside the material with time. The K-V model’s constitutive law shows that for
de/dt = 0, stress would be constant for all time (and equal to exactly Ee), and never relax; a real polymer does relax
over time, so K-V poorly predicts polymer relaxation.
9. Graph of stress-strain curve showing the energy loss and the concept of mechanical damping. Can you
please explain the aeroplane explanation concept again?
If you load a linear elastic material, the strain in the material would instantaneously be equal to the stress/E, no
matter how fast or slow your loading frequency. In contrast, if you stress a polymer at time = 1/frequency much
faster than its characteristic time (τ = η/E for the Maxwell or Kelvin-Voigt models) or much slower than its
characteristic time, you’ll observe the strain that you would predict from the constitutive models for short times (t =
0) or long time (t = infinity), respectively. If you stress a polymer at 1/frequency that is intermediate to these two
extremes, the “viscous” or “dashpot” nature of the polymer will cause the strain to lag behind your loading frequency.
The polymer is damping the input mechanical stress via this lag, often dissipated or as heat. You would see this in a
stress-strain curve as a hysteresis, or gap between the loading part and the unloading part, where the area inside
that hysteresis loop is the dissipated energy. The example I gave in class was actually of a tire material (a
reinforced rubber, not really a good example of a LVE polymer governed by internal energy U) where the goal is to
design the material to damp dissipate as much mechanical energy as possible at the operating frequency, which
might be the angular frequency of an airplane or truck or car tire.
10. Are we responsible for the fine details of the derivations of the force and entropic parameters that
characterize the polymer chains? Gaussian distribution function?
No, you do not need to memorize the form of a Gaussian distribution function. Yes, you should understand and be
comfortable deriving the molecular origin of rubber elasticity and predicting the effects of changing Lp, Lc, T, etc on
entropic spring constant. From previous Course 3 classes, you can relate the Hemholtz energy in terms of entropy
as a function of accessible configurations that can be expressed by some probability distribution function.
11. Can you please address the main differences between the tresca and Von Mises criteria? Also can you
please discuss the concept of conservatism?
Tresca: material yields when max shear stress = max shear stress attained when material yields under uniaxial
loading; this neglects the intermediate principal stress.
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von Mises: material yields when the shear strain energy reaches a critical value, which can be expressed in terms
of the effective stress σeff = effective stress attained when material yields under uniaxial loading; this considers all 3
principal (normal) stresses.
Conservative: you are conservative if you predict yielding at stresses that could be lower than the actual stresses
required for yielding. This is a conservative estimate of when yielding will occur, because those stresses may
actually be insufficient to cause yielding. If you want to ensure that yielding does not occur under some triaxial
stress state, you typically want to err on the conservative side of your estimates so that you’re quite sure yielding
will not occur.
12. I remember writing in my lectures notes that yielding is defined as the stress needed to increase
dislocations or cause the dislocations to multiply(during the guest lectures)-I may have written the point
incorrectly. Can you please explain? In general, is it safe to assume that having dislocations in a material is
advantageous because it provides strengthening mechanisms in the material and increase the yield
strength of the material? That said, can you please emphasize the role of dislocations in plastic
deformation/yielding?
Yielding means “plastically or permanently deforming without fracturing/breaking”. The motion of dislocations is the
microscopic origin of plasticity; dislocations are the carriers of plasticity in a crystal, just as electrons are the carriers
of current. At the microscopic level, this happens as soon as the first dislocation moves. From an engineering point
of view, motion of just one dislocation is usually inconsequential, so we define “yield stress” as the stress we can
measure beyond which the material is plastically deformed. To measure such a stress practically, the
displacements must be much greater than a single Burgers vector due to one dislocation. In fact, we need lots of
dislocations to move, so we say that yielding is measureable when lots of dislocations move, which means the
stress is sufficient to move existing and generate new dislocations.
Will having dislocations always increase the yield strength of a crystal? This is a tricky concept: as we discussed, a
single crystal with zero dislocations will have a higher yield strength than a single crystal with some pre-existing
dislocations; to yield the former case, we’d have to nucleate some dislocations at GPa-level stresses. On the other
hand, a single or polycrystal with 1000 pre-existing dislocations would generally have a higher yield strength than
one with 100 pre-existing dislocations, for the reasons related to work hardening that we discussed.
13. What is the relationship between the pinning points and plasticity? I am under the impression that the
pinning points prevent plasticity from taking place by impeding the motion of the dislocations.
Pinning points “pin” segments of a moving dislocation, such that it takes additional stress to continue to move that
segment of the dislocation (it must get longer to curve between those pinning points, and its energetic penalty
increases per unit length of the dislocation). You are correct: since plasticity requires motion of dislocations, pinning
impedes dislocation motion and thus increases the stress required for sustained plastic deformation. However, a
point beyond what we’ve covered in class in detail (and will cover in 3.14) is that these pinning points also can
serve as sites of dislocation multiplication: looking at the precipitate/particle strengthening, we see that when
dislocations are bent sufficiently around an obstacle, parts of the dislocation “pinch off” and form new “child”
dislocations that can move independently of the “parent” dislocation.
14. Do we need to qualitatively explain the strengthening mechanisms, i.e alloying elements, dislocations
and grain boundaries or are supposed to quantitatively describe these concepts?
Both: You need to understand the physical explanation for why each of these approaches increases the yield
strength, and quantitatively know how a change in the microstructure will affect the yield strength of the material.
These equations are quite simple because they are based on empirical (experimental) observations, but are good
engineering tools for material design.
15. The general concept that I am the most confused about is the worm-like chain model and
the freely-jointed chain model, and the various parameters we can derive from these
models.
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FJC model: Force required to extend this chain depends on contour length Lc, which depends on the number and
length of rigid segments that make up the chain, n and b, respectively.
WLC model: Force required to extend this chain depends on Lc and Lp. Here, Lp is like b from the FJC model, but
reflects the fact that polymer chains are not made up of rigid segments b connected by joints, but flexible segments
that fluctuate independently over some length Lp.
Both models are based on entropically dominated free energy, where entropy is defined by accessible
configurations of the chain as derived in class. As in pset 5, one can apply these models to experimentally
measured data to estimate Lc, Lp, and/or n and b of the chain, and the stiffness of that polymer chain.
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